Download Section 17

Equations of motion of a charge
in a field
Section 17
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The field acts on a charge e
The charge e acts on the field
But if e is small, we can neglect the 2nd effect
Then the field is independent of position and velocity of e.
Hamilton’s principal varies the action. The
condition for a minimum determines the
equation of motion.
r and v are the usual coordinates and velocity of
the particle,
But
is not the usual particle momentum.
It is the generalized momentum.
There is an extra term due to the A.v in L.
Lagrange
equation of
motion
This gradient acts on the r dependence of A and f.
(v is considered an independent variable.)
HW
In this problem
The derivatives are taken
at constant v.
Equation of motion for particle in given fields
Equation of motion has the total time derivative of A.
The total differential of A is
found using the chain rule.
Change of A at fixed r.
Change of A due to motion
from r to r + dr at fixed time.
For example, suppose
Divide dA by dt
Sub into the equation of motion:
These two terms are the same and cancel.
Equation of motion of charge in given field F = dp/dt
Independent of v.
Proportional and
perpendicular to v.
Electric field.
A polar vector
Magnetic field.
An axial vector
(Doesn’t change sign under inversion.)
Relativistically correct “Lorentz force”
Low velocities
Rate of change of kinetic + rest energy , ekin
HW
Work
equation
Work done by the field on the
particle per unit time (Power)
The magnetic field does no work on the charge since the
magnetic part of the force is perpendicular to dr
Time reversal symmetry
• Equations of mechanics are invariant under time reversal.
• Relativistic equations of motion in E-M field are unchanged by the
transform
• Then
The effect of the time-reversal transform on the potentials is
If a certain motion is possible in an EM field, the reversed
motion is possible in an EM field having H reversed.
Motion in a magnetic field.
• H
F
F
X H
F
Reversed motion
with reversed
magnetic field
• H
If you don’t reverse the Hfield, the motion is not the
reverse.
What effect acting alone causes no change in a particle’s kinetic
energy?
• A spatially varying vector potential
• A time dependent vector potential
• A spatially varying scalar potential
What effect acting alone causes no
change in a particle’s kinetic energy?
• A spatially varying vector potential
• A time dependent vector potential
• A spatially varying scalar potential